829
ISDIRECT COLORIMETRY SUMMARY
The general equations for radial flow adsorption columns are derived and their solution obtained for two different cases. The first case is that in which equilibrium is established. The solution is obtained by means of the La Place transform for the case of a linear isotherm. This method is not of use for nonlinear isotherms. The second case covers the nonequilibrium situation, and in it the authors assume a kinetic equation of the Langmuir type. General solutions are obtained for this case. KOexperimental confirmation of these equations is presented but work is proceeding in this direction. REFERENCES (1) AYUNDSON, N . R . : J. Phys. B- Colloid Chem. 62, 1153 (1948). ( 2 ) British patent 611,080 (October 25, 1948). (3) CHURCHILL, R . V. : .Vodern Operational Mathematics in Engineering. McGraw-Hill Book Company, Inc., Kew York (1944). (4) HOPF,P. P.: Ind. Eng. Chem. 39, 938 (1947). (5) MEINHARD, J. E . , A N D HALL,N. F.: Anal. Chem. 21, 185 (1949). (6) SASBORN, C. E., AND.AMCNDSOX, N. R . : .4m. Math. Monthly 66, 217 (1949). (7) T ~ o v a H. s C.: Ann. S . Y. Acad. Sci. 49, 161 (1948). (8) TRUEBLOOD, K. K.:“An Experimental Study of Chromatography on Silicic AcidCelite”, in press. (9) WILLIAMS, T. I . : Introduction to Chromatography. Blackie and Son, Glasgow (1947). (10) WILSON,J. S . : J. .4m. Chem. SOC.62, 1583 (1940).
INDIRECT COLORIMETRY
THE DETERMIK.4TION OF
O F THE APPARENT DISSOCIATION CONSTANTS MOiYONITROQUINOL, 2,6-DIiVTROQUII\’OL,AND 3-ALIZARINS~LFO~IC ACID
LUIGI SACCONI Institute of Physical Chemistry, Unzuersity of Florence, Florence, I t a l y Received February 66,1949 THEORY
I t is Fell knoua that if two colored substances, A and B, each absorbing light independently, are present in a homogeneous system (e.g., solution) the ratio cA/cB of their concentrations can be determined colorimetrically if the following conditions are fulfilled: (a) the total stoichiometric concentration is known; ( b ) both A and B can be isolated in order to prepare two standard solutions for comparison with solutions containing the substances in an unknown ratio cA,’cB. This last condition is not always satisfied. I n a system of the type: p h f nX f
... + p B
f TY
+ ...
(1)
830
LUIGI SACCONI
where A and B are the only colored components, the equilibrium cannot alivays be completely displaced to one side or the other. Moreover, in the attempt to isolate either A or B, reactions may occur which alter the system itself. The object of this study is to point out that, even vhen it is not possible or convenient to isolate the components A and B, the problem can be solved under the follop4ng condition: the ratio of the stoichiometric concentrations of the two components must be a function of a variable a which may be directly measured, or evaluated in terms of other knoivn quantities characteristic of the system. In other words, for any ratio, c,/c, we must have
where f represents an unknoivn function (otherwise the c A / c B value could be directly determined by calculation). One typical system where such a condition holds is the solution of an acid-base indicator, the equilibrium of which may be represented as follows: H I 2 H + I-. In fact, the system is described in general by a function F which relates the ratio of the concentrations of both H I and Ito the reciprocal of the hydrogen-ion activity according to the following equation:
+
%=.(A)
CHI
(3)
in which the function F represents the apparent dissociation constant KI, constant for constant ionic strength and defined by the well-known relation:
In the case of an indicator, it is often impossible to isolate both the optical
HI and I- forms. To this the solution must usually be brought to a high alkalinity or acidity, which may cause either the decomposition of the indicator or give rise t o a salt error. Moreover, for an indicator behaving as a dibasic acid according to the scheme: HzI e H f HI- e 2H’ I--, the two color changes may be superposed. In the latter case, the intermediate form HI-, for any given pH value, never exists alone in the solution, and hence cannot be isolated to prepare the standard solution of the pure HI- form. In these circumstances it is impossible to determine the value of cI-/cHIby the usual colorimetric methods, or, for a double color change, the values of C ~ I - , ’ C H ~ I and cI--/cAI-, as well as the value of pK, or for a double color change, of pK1 and pK2. By the following method of indirect colorimetry, all these determinations can be carried out.
+
+
METHOD O F DOUBLE INCOMPLETE COLOR CHANGE
Case I: Neither of the two forms H I and I- o j the indicator can be isolated. When three values of the hydrogen-ion act,ivity aiH+,azH+, aaH+,are known
831
ISDIRECT COLORIJIETRP
for three different equilibrium conditions of the system, the following equations hold :
This indeterminate system of three equations with four unknowns can become determinate in the folloning manner. Suppose that a colorimeter (figure 1) is formed by two joined wedge cells 1 and 2 and a movable cell 3, and suppose that the solutions of the indicator contained in the three cells have the same concentrations. Let alH+> a%=+> ma+be, respectively, the hydrogen-ion activities of the three solutions, and ciI-,lciHI, czI-:'~zHI, be, respectively, the ratios of the concentrations of the two forms in t'he three solutions; then the dissociation constant KIwill be defined by relation 4. S o w let the cell 3 be adjusted over cells 1 and 2 until its color matches the color resulting from the f1T-o joined cells. I n figure 1, X N represents the optical path through the cells. Suppose the two forms H I and I are separated in cells 1 and 2 by two imaginary planes (the traces of which are dotted in figure 1j d r a m
FIG 1 FIG.2 FIG.1. Scheme of the double incomplete color change method FIG 2 Scheme of the simple incomplete color change method from the corners d and C, and so arranged as to delimit in each cell two zones in which the respective concentrations of the HI and I- forms are equal: the segment MW will cut the t x o traces of these planes and the trace of the boundary wall between the two wedge cells in the points P , S, Q . If MQ = b , M P = x, QN = c, S S = y, then: Y - c x . b -x c -y CZHI' c -y On substituting these values in equations 4 we find:
b - x = z . __ 5
ClHI
'
+ Y = cx
+x
CsHI
(5)
S o w , b and c are given by colorimetric measurements, and if mH+, mH+ are known, the system of three equations with three unknown quantities (KI,z, y) can be solved By dividing the members of the equations 6 by ulH+ and putting uzH+ a i H d = h2, ~ 3 ~ + , ' a=i ~ha. + Tve find:
832
LUIGI SACCONI
From the first equation we obtain:
-
c(b X) hzx+ b - x On substituting the value of y in equation 7b, rearranging, and simplifying, we find: hzhsx' - hsx2 - hzxZ X' hsbx - ~ Z C X h 3 ~ bx ~ =0 hence, dividing by x and solving with respect to this unknown quantity we obtain : x = b hzc h3(b C) (9) hZh3 - (hZ h3) 1 Substitution of x in equation 8 gives y, and from equation 5 we obtain the value of the three unknown ratios cI-/cHI = a j ( 1 - a) for the three solutions in the three cells. By substituting the x- and y-values in one of equations 6, the values of KI and pK are finally obtained. For example, from Y =
+ +
+ -
by putting pH8 =
-
+
+ + +
KI = asH+'b - x + y c+x-y log as=+,we obtain:
Case IZ: In the caseof the equilibrium ina buffer-indicator system of the type:
I-
+ HA e H I + A-, which defines the concentration equilibrium constant (8): K -
cI- ' %A
CHI'CA-
which is constant for a constant ionic strength, being cI-/cHI = F(c~-/cHA), it is, similarly, possible to determine by indirect colorimetry the values of the ratio cI-/cHI and the K-value, even in the case when neither of the I- and H I forms can be isolated. In fact, if three values of the ratio C H A j C A - are known, we can write the system of equations:
K - CII-'CIHA C1HI'CLA-
CzI-'CzHA CzHI'cSA-
- C31-'CaHA CaHI'CSA-
Assuming the above explanation and setting C i H A j C i A - = R1, c'HA/c'AC s H A j C s A - = &, Rz/R1 = r2, &/R1 = ~ g we , obtain in a similar way:
x= b
+ rzc - r d b + c)
T Z T ~- ( T I
+ + 1' r3)
c(b = TZX
=
Rz,
- X)
+b -x
and if x and y are known, the three ratios cI-/cHI can be determined, as well as the value of K , from equation 5 and from the following: C - Y
I N D I R E C T COLORIMETRY
833
METHOD O F S I M P L E I S C O M P L E T E COLOR C H A S G E
Case I : Only the acid form HI can be isolated. Let the acid solution be placed in the cell 0 (figure 2) and a solution of hydrogen-ion activity a i H + be placed in cell 1. Let the ratio of the alkaline to the acid form in this solution be ciI-/cim and the corresponding values for the solution in the cell 2 be respectively UZH+ and C Z I - / C Z ~ (azH+ ~ > aiH+).The resulting system of two equations with three unknown quantities:
is indeterminate. E
C
20
20
I5
10
10
5 a,..
o
D
A
FIG.3 FIG. 4 FIG.3. Scheme of the simple incomplete color change method FIG.4. Graphic verification for 3-alizarinsulfonic acid
If we find the match between cell 2 and the joined wedge cells 0 and 1, and mark the trace of the imaginary boundary plane between the two I- and HI forms in the cell 1, the above treatment mill give: c1I-/clHI
=
SS/SQj
Setting MJV = d , QN= b, KV = 10, we obtain:
C~I-/CZHI = SAV/MS
2,QS =
b
- x , and substituting in equations
when ai=+,aiH+,and d (thickness of the solution layers) are known, the measurement of b makes the system determinate. For, putting azH+jalH+= q, from equation 11 we have:
Case 11: Only the alkaline form can be isolated. Let its solution be placed in the cell 0 (figure 3), and two solutions Tvith hydrogen-ion activities aiH+and azH+ (ai,+ > azH+),respectively, be placed in cells 1 and 2. If the ratios between the
834
LUIQI SACCONI
two forms in the solutions are treatment gives: C1I-/Clm
= Q S / S N = (b -
ClI-/c1HI,
czI-/cznr, respectively, then the above
x)/x;
c ~ - / c ~ H I=
M S / S N = (d
- X)/X
If aiH+/aza+= Q:
x
Qb
-d
= - ,
KI
=
airqt.-
d-b
ob - d
Thus by the double incomplete color change method, in place of two standard solutions, each containing one of the two pure colored forms, two fictitious standard solutions, included within the range of the color change, are employed. In the simple incomplete color change method, when only one optical form can be isolated, then in place of the standard solution of the form which cannot be isolated, a Jictitious standard, within the range of the color change, is employed. I t could also be shown that the indirect method can be applied with a Gillespie colorimeter (4).The validity and accuracy of the method have been tested and established in previous researches (14). EXPERIMENTAL
The wedge colorimeter used was of the Bjerrum-Kolthoff type (2). The upper field of the movable cell and the lower of the joined wedge cells are brought to optical contiguity by a prism arrangement. In the calculations, instead of the whole thickness of the two absorbing layers in the two joined cells, the breadth A D of the cells is used ( A D = 21.20 cm.; 21.35 cm). In place of b- and c-values, corresponding to the parts ir. which the segment MLV is divided by the wall A C , as indicated in the previous scheme, the proportional values N D and A N are used. The scale A D is graduated in 0.5 mm. Combinations of blue and green filters were used to make the transmitted light as nearly as possible a neutral grey. McIlvaine’s buffer solutions (10) (0.1 M citric acid and 0.2 M disodium hydrogen phosphate) were used for the first color change of the dinitroquinol and 3-alizarinsulfonic acid. A Kolthoff buffer (6) vas used for the first color change of mononitroquinol. For the second color change of the mono- and di-nitroquinols 0.1 M sodium carbonate 0.1 ,If boric acid 0.1 N potassium chloride buffers (7) were used. For the second color change of the Alizarin S Ringer buffers (9) (0.15 M disodium hydrogen phosphate and 0.1 M sodium hydroxide) were used. I t is assumed that the difference in the observed values caused by inconstant ionic strength falls within the experimental errors. The mononitroquinol was prepared according to Richter (13), and was recrystallized to constant melting point (133°C.). The dinitroquinol was prepared according to Nietzki (11) and recrystallized t o constant melting point (131136°C.). Alizarin S from Grubler and Co., Leipzig, was used. The indicator solutions used were: mono- and di-nitroquinols, 0.1 per cent in 20 per cent alco80 per cent water mixture; Alizarin S, 0.1 per cent solution in water. hol
+
+
+
ISDIRECT COLORIMETRY
835
D I S C T X X O S , PROCEDURE, .ZSD RESULTS
The mono- and di-nitroquinols, in strong acid solutions, show a greenish yellon color. The color of mononitroquinol turns to orange within the 5.5-9.2 pH range. and t o violet within the 9.2-11.2 pH range. For the 2.6-dinitroquinol (5) the color change toward> orange takes place xithin the 3.0-5.8 pH range. and the orange to violet color change lvithin the 8.0-10.0 pH range. The electrometric determinations of pK values of these substances gave nonreproducible values because of decomposition (3). Prideaux and S u n n (12), in a colorimetric method, assumed the existence of three optical forms, yellow, orange, violet (in addition to the acid one), corresponding to three colorimetric constants, the values of Irhich are found to be: for mononitroquinol pKl = 3.25, pk': = 7.2. pK3 = 10.2;for 2 .G-dinitroquinol pKl = 2.8, pK2 = 5.2, pKa = 9.05, The yellow form n a s not actually isolated to prepare its corresponding standard solutions, but the alkaline form of dinitrophenol n-as arbitrarily employed in its place. Further, the color changes being multiple and probably superimposed. the existence of the pure orange form in the orange standard solutions used 11-as not assumed, as it was not demonstrated that some solutions used as standard (e.g.,rnononitroquinol at pH = 5 . 5 , at the beginning of the rolor change towards orange) contain only one pure form. Since the violet form of both indicators, existing in a strongly allialine solution, easily decomposes. the standards of the pure alkaline form are unstable. I t is thought that indirect colorimetry is suitable for the determination of the pk' values of these substmces. DETERJIISATIOX O F PK1 O F THE M O S O - A S D DI-XITROQUISOLS
Jf ononiiroquinol
A standard solution, containing the pure greenish yellow (acid) form of the substance (Solution 0, 0.1 A' hydrochloric acid) \vas prepared. As a fictitious orange standard, a series of solutions was used of such pH that the color change toivards orange could not be fully achieved (Solution 1, pH1 = 7.60-7.80-8.0). Suitable intermediate solutions (Solution 2, pH2 = 7.00-7.20-7.40-7. 60) were used. Solution 2 was matched xith 0 and 1 conjointly so that the ~ ' ( -b X ) = '(1 - a),K~ =
z values n-ere found. The fact that the pK values b - x
found are, within the experimental errors, close to the same value (7.63) shows that the color change from greenish yellolv to orange is simple (see table 1). In fact, if the indicator had two different greenish yello\v-yellow and yellonorange color changes, the fictitious standard Solution 2 would result as a mixture of the two forms, the y e l l o ~and the orange one. Then for each p H value of Solution 2, a different pK value would be obtained, situated between the value of a pk' corresponding to the greenish yellovyellon color change and t,he value of a pK corresponding to a yellon-orange color change. By the double incomalete color change method (see table 2) it has been definitely demonstrated that the color change from greenish yellow to orange is
836
LUIGI SACCONI
simple. As a fictitious acid standard, a solution with a m-eak but definite concentration of the orange form was chosen, viz., a solution the color of which lies a t the beginning of the color change from greenish yellow to orange (or from yellow to orange, if the yellow form exists) (Solution I, pH1 = 6.80). As a fictitious orange standard, a series of solutions was used with such a pH (Solution 2, pHs = 8.20-8.40) as to allow the larger part of the indicator to exist in the orange form and the smaller in the acid one (greenish yellow or yellow). Intermediate solutions (Solution 3, pH3 = 7.40-7.80) were employed. Since the TABLE 1 p K I value of mononitroquinol determined b y the simple incomplete color change method Stoichiometric concentration of indicator = 1.03 X T = 21.0"C.
__ PHI
___ ZH+ x 1(
pH1
'E+X l o B l
~
7.60 8.00 8.00 8.00 8.00 7.80 7.80
1.00 1.58 1 58
Mer
value
-
1 .ooo 3.98 2.51 0.251 3.98 0.396 6.31 0.631 1.ooo 10.00 1 .ooo 6.31 3.98 0.631
100
~
,
PKI
Y
__
~
7.00 7.00 7.20
__
1
8.32 14.51 11.44 8.16 5.90 6.73 9.43
-
~
21.20 21.20 21.20 21.20 21.20 21.20 21.20
I ~
'
~
I
3.99 10.07 8.16 5.71 4.20 4.00 6.48
48.0 69.4 71.3 70.0 71.2 59.4 58.1
7.63 7.64 7.61 7.64 7.61 7.63 7.66
__
7.63f0.02
TABLE 2 p K , value of mononitroquinol determined b y the double incomplete color change method T = 21.0"C. Stoichiometric concentration of indicator = 1.03 X x 6.80 1.58 8.20 6.80 1.58 8.20 ~
3.98 6.31 6.31 6.31 3.98
1 7.60 7.60 ~
I
C
108,
2.51 2.51 7.40 3.98 7.80 1.58 7.80 1.58
-l
I ~
11.16 9.89 13.79 6.25 8.00
Mean value. , . . . . . . . , . . . . , . . . . . . . . . . . . . . .
10.04 9.82 11.31 8.64 7.41 12.21 5.47 14.95 7.08 , 13.20 ,
, ,, ,
8.47 0.862 8.87 0.913 6.21 0.581 11.69 1.430 11.06 1.301
7.66 7.64 7.64 7.65 7.68
I'
. . , , , . . . . , , . . . , , , . . . . . . 7 65 =t0.02
orange form always exists in the two fictitious standards, the solutions will always remain in the range of the greenish yellow-orange or within the range of the yellow-orange color change, if the yellow form exists. In the last case the determined constant should correspond to a yellow-orange color change, and therefore it should be different from that previously determined for the color change from greenish yellow to orange. Actually, the pK1 value found is identical, within the experimental errors, with that determined by the simple incomplete color change method. We therefore conclude that the color change from greenish yellow to orange is simple and that a yellow form does not exist.
837
IKDIRECT COLORIMETRY
8 -
- 8
4.80 4.80 5.20 4.40
1 ~
~
!
-I-
~
1.58 4.40 1.58 1 3.80 0.63 ' 4.20 3.98 3.60
i
~
0.398 1.580
~
2.51 10.00 10.00 6.31
1 ::::: I '
,
1 ~
14.63 5.47 9.44 5.69
~
i
21.20 21.20 21.20 21.20
,
'
1
10.28 3.72 8.13 2.77
i
1 1
70.3 68.1 86.2 48.6
4.43 4.43 4.41 4.42
TABLE 4 pK1 value o f 2,6-dinzlroqutnol defermzned by the simple ancornplele color change method Stoichiometric concentration of indicator = 0 8 X lo-', T = 21 0°C
__
~
3 40 3 40 3 40
3 98 3 98 3 98
I
-~
4 20 4 40 4 41
LIenn balue
4 43 f 0 03
The pKl values found are the same in both series (see tables 3 and 4); hence it is concluded that a yellow form of 2,6-dinitroquinol does not exist, and therefore the greenish yellow-orange color change is simple. Both the nitro compounds behave as dibasic acids, following the scheme: H J G H+ HI- G 2H' I--, where H J corresponds to the greenish yellow form, HI- t o the orange, and I-- to the violet. The indirect method offers a control which makes it possible, in the case of multiple color change, to avoid using, as a standard, a solution containing also a form situated outside the investigated color change (e.g., the violet form which may be present in the standard orange solution). In fact, using filters of a color complementary to that leaving the colorimeter, it can easily be seen if the color
+
+
838
LUIGI SACCONI
of the intermediate solution fails to match the color from the solutions of the joined wedge cells. DETERMIXATION OF
PKz VALUES
OF MONONITROQUINOL AND
2,6-DINITROQUINOL
Mononitropuinol The double incomplete color change method (see table 5 ) was used for the orange-violet color change, to avoid a strong alkalinity causing the decomposition of the only existent violet form. For this reason, a fictitious violet (alkaline) standard at pH2 = 10.80 (Solution 2) such that the color change toward violet should not be complete was employed. Since the contiguity of the greenish yelloworange color change to the orange-violet one does not allow the rigorous isolation of the pure orange form, a fictitious orange standard solution (Solution 1, pHl = 9.20) was used in place of an orange standard. This pH is within the range of the orange-violet color change; hence this solution contains only the orange and violet forms. The pH values of the intermediate solution (PHI, Solution 3) are 9.00-10.00-10.20. TABLE 5 pK? value of mononitroquinol determined b y the double incomplete color change method Stoichiometric concentration of indicator = 0.64 X lo-’; T = 21.0”C.
____ 6 04 0 571 8 38 0 843 11 14 1 323 Mean value. .
.
..
.,
. .. ... .. . . . . ..,
...
.
10 04 10 07 10 08
.. . . .
I t can be seen that pK2 - pK1 = 10.06 - 7.64 = 2.42. According to the general equation for indicator equilibrium, pH - pK = log a / ( 1 - or), for a pH = 8.85, halfway between 7.64 and 10.06,we obtain: 8.85 - 7.64 = log a’/(l - a’),1 - or’ being the fraction of indicator which has not been transformed into the orange form = 1/17; 10.06 - 8.85 = log (1 - W ” ) / ( Y ” , a” being the fraction of the indicator transformed into the violet form = 1/17, respectively, for the first and the second color changes. We can therefore assume that an orange solution with pH = 8.85 contains 1/17 of both greenish yellow and violet forms. In this case, it is demonstrated that the ranges of both color changes are superimposed and that the standard solution of the pure orange form cannot be prepared. If we calculate the amount of the violet form in the fictitious orange standard at pH = 9.20, above employed, we find that a” = ciI--/(cil-cid = ( b - x)/b = 10.5 per cent, and the concentration of the orange form must be 89.5 per cent. I t is thus confirmed that the solution a t pH = 9.5, used in the investigations reported (12) as standard of the pure orange form, does not contain only this last form. In fact, at pH = 9.5 the concentration of the violet form is certainly larger than that which must exist a t pH = 9.20 (10.5 per cent).
+
839
ISDIRECT COLORIMETRY
2,6-Dinitroqiiinol For the orange-violet color change which ranges far from the greenish yelloworange one, an actual orange standard (pH = 6.80-6.90) is employed. To avoid decomposition of the substance in strongly alkaline solution, a series of fictitious standards (Solution 1, pH1 = 9.20-9.40-Y.60) is used. These solutions, where both orange and violet forms exist, are stable enough to allow a series of measurements (see table 6). The pHz values of the intermediate solutions (Solution 2) are 8 60-8.80-9.00. Summarizing : Mononitroquinol: pKi = 7.64 =!z 0.03; pK2 = 10.06 =k 0.02 2,6-Dinitroquinol: pK1 = 4.43 & 0.03; pKz = 9.14 + 0.03 TABLE 6 p K I zalue of 2,6-dinitroquinol determined by the simple incomplete color change method Stoichiometric concentration of indicator = 0.5 X lo-?; 2' = 2 1 . 0 T . I
pHi
_
'''EC
x
pHi
1010
lo*E+x109
;x
p
_ __ ~
9 6 0 ~2 5 1 9 6 0 ' 251 9 40 3 98 9 40 3 98 920 631 9 20 6 31
' '
1
1
8.60 8.80 9.00 8.80 8.60 8.80
I
2.51 1.58 1.00
~
~
10.00 6.31 2.51
6 8 14 10 8 12
16 98 12 06
95 25
i 1 '
21 21 21 21
20 20
-
~
'
~
20
21 20 21 20
72.8 6 6i i5.5 9 43 66.7 6 32 62.8 4 84 , 5 L l 6 32 51.6 -I49
20
~
I
1
1
_
Mean value
pKz
100
_
,
9.17 9.14
1
9.10
1 '
9.17
9.13 9.17
~
9 14
D E T E R ~ f I S . ~ T I OO~F THE PK1 A S D
PKz
VALUES O F
=to 03
3-ALIZ.%RIXSCLFONIC .ICID
This substance, the sodium salt of which is called Alizarin Red S, shows in strong acid solutions a greenish yellow color. K i t h decreasing hydrogen-ion concentration the color turns hroivn, reaching violet, through red and lilac, in strongly alkaline solution. A. I
